What is the square root of `((x^(5)-1)/(x-1))+(x^(3)+2x^(2)+x)` ?

Options

A. `x^(2)+x+1`
B. `x^(2)-x+1`
C. `x^(2)-x-1`
D. `x^(2)+x-1`

Answer

To solve the problem of finding the square root of \(\frac{x^5 - 1}{x - 1} + (x^3 + 2x^2 + x)\), we can follow these steps:

### Step 1: Simplify the expression \(\frac{x^5 - 1}{x - 1}\)

We start with the expression \(\frac{x^5 - 1}{x - 1}\). We can factor \(x^5 - 1\) using the difference of cubes formula:

\[ x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1) \]

Thus, we have:

\[ \frac{x^5 - 1}{x - 1} = x^4 + x^3 + x^2 + x + 1 \]

### Step 2: Combine the expression with \(x^3 + 2x^2 + x\)

Now, we add the second part of the expression:

\[ x^4 + x^3 + x^2 + x + 1 + (x^3 + 2x^2 + x) \]

Combining like terms:

\[ x^4 + (x^3 + x^3) + (x^2 + 2x^2) + (x + x) + 1 = x^4 + 2x^3 + 3x^2 + 2x + 1 \]

### Step 3: Recognize the expression as a perfect square

Next, we need to check if the expression \(x^4 + 2x^3 + 3x^2 + 2x + 1\) can be expressed as a perfect square.

Notice that:

\[ x^4 + 2x^3 + x^2 + 2x + 1 = (x^2 + x + 1)^2 \]

Thus, we can rewrite our expression as:

\[ (x^2 + x + 1)^2 + 2x^2 \]

### Step 4: Combine terms to form a complete square

We can further simplify:

\[ (x^2 + x + 1)^2 \]

### Step 5: Take the square root

Finally, we take the square root of the entire expression:

\[ \sqrt{(x^2 + x + 1)^2} = x^2 + x + 1 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{x^2 + x + 1} \]